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adm_strat
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[SOLVED] Surface Area
Find the area of the surface of the part of the plane x + 2y + z = 4 that lies inside of the cylinder [tex]x^{2} + y^{2}=4[/tex]
[tex]A(S)= \int\int_{D} \sqrt{1+( \frac{\partial z}{\partial x})^{2} + +( \frac{\partial z}{\partial y})^{2}} dA [/tex]
I can tell intuitively that the intersection is a ellipse. When I set the two equations equal to each other I get the equation:
[tex]z= x^{2}-x+y^{2}-2y[/tex]
I am having a brain fart and can't seem to do the double integral. Sorry, its finals week. I know that I need to do a change of variables, but I don't know what since it is an ellipse in [tex]R^{3}[/tex] I can complete the square, but it didn't seem to lead me anywhere useful.
Any help would be appreciated. Thanks in advance.
Homework Statement
Find the area of the surface of the part of the plane x + 2y + z = 4 that lies inside of the cylinder [tex]x^{2} + y^{2}=4[/tex]
Homework Equations
[tex]A(S)= \int\int_{D} \sqrt{1+( \frac{\partial z}{\partial x})^{2} + +( \frac{\partial z}{\partial y})^{2}} dA [/tex]
The Attempt at a Solution
I can tell intuitively that the intersection is a ellipse. When I set the two equations equal to each other I get the equation:
[tex]z= x^{2}-x+y^{2}-2y[/tex]
I am having a brain fart and can't seem to do the double integral. Sorry, its finals week. I know that I need to do a change of variables, but I don't know what since it is an ellipse in [tex]R^{3}[/tex] I can complete the square, but it didn't seem to lead me anywhere useful.
Any help would be appreciated. Thanks in advance.
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